integral of ((x+3))/(x^2+11x+18)

Lösung

\int \frac{( x + 3 )}{x ^{2} + 11 x + 18} d x

\frac{1}{7} ln ∣ x + 2 ∣ + \frac{6}{7} ln ∣ x + 9 ∣ + C

Schritte zur Lösung

\int \frac{x + 3}{x ^{2} + 11 x + 18} d x

Wende U-Substitution an

= \int \frac{u}{u ^{2} + 5 u - 6} d u

Ermittle den Partialbruch von

\frac{u}{u ^{2} + 5 u - 6} : \frac{1}{7 ( u - 1 )} + \frac{6}{7 ( u + 6 )}

= \int \frac{1}{7 ( u - 1 )} + \frac{6}{7 ( u + 6 )} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{7 ( u - 1 )} d u + \int \frac{6}{7 ( u + 6 )} d u

\int \frac{1}{7 ( u - 1 )} d u = \frac{1}{7} ln ∣ u - 1 ∣

\int \frac{6}{7 ( u + 6 )} d u = \frac{6}{7} ln ∣ u + 6 ∣

= \frac{1}{7} ln ∣ u - 1 ∣ + \frac{6}{7} ln ∣ u + 6 ∣

Setze in

u = x + 3

ein

= \frac{1}{7} ln ∣ x + 3 - 1 ∣ + \frac{6}{7} ln ∣ x + 3 + 6 ∣

Vereinfache

\frac{1}{7} ln ∣ x + 3 - 1 ∣ + \frac{6}{7} ln ∣ x + 3 + 6 ∣ : \frac{1}{7} ln ∣ x + 2 ∣ + \frac{6}{7} ln ∣ x + 9 ∣

= \frac{1}{7} ln ∣ x + 2 ∣ + \frac{6}{7} ln ∣ x + 9 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{7} ln ∣ x + 2 ∣ + \frac{6}{7} ln ∣ x + 9 ∣ + C

\frac{d}{d y} (y e^{2 x y} + x)

d er i v a t i v e y = ln (4 x)

\int \frac{- 2}{x ^{2}} d x

\frac{d y}{d x} = \frac{x}{y ^{3}}

\frac{d y}{d t} + \frac{t}{t ^{2} - 9} y = \frac{e ^{t}}{t - 6}